We show that the Cabibbo angle theta\_c satisfies the relation tan(2
theta\_c)=-+/- 1/2 when universality for diagonal neutral currents of mass
eigenstates is satisfied at the same level of accuracy as the absence of their
off-diagonal counterparts. The predicted value is cos theta\_c \approx 0.9732,
only 7/10000 away from experimental results. No mass ratio appears in the
calculation. theta\_c occurs a priori for both quark species, and, showing that
one recovers the standard dependence of leptonic and semi-leptonic decays of
pseudoscalar mesons, we advocate, like for neutrinos and charged leptons, for a
symmetrical treatment of u and d-type quarks.Comment: The Cabibbo angle can be positive or negative, which was overlooked
  in the first version. Minor addend

Duret, Quentin

Machet, Bruno

English

arXiv

The Cabibbo angle can be positive or negative, which was overlooked in the first version. Minor addenda.We show that the Cabibbo angle theta_c satisfies the relation tan(2 theta_c)=-+/- 1/2 when universality for diagonal neutral currents of mass eigenstates is satisfied at the same level of accuracy as the absence of their off-diagonal counterparts. The predicted value is cos theta_c \approx 0.9732, only 7/10000 away from experimental results. No mass ratio appears in the calculation. theta_c occurs a priori for both quark species, and, showing that one recovers the standard dependence of leptonic and semi-leptonic decays of pseudoscalar mesons, we advocate, like for neutrinos and charged leptons, for a symmetrical treatment of u and d-type quarks

Hal-Diderot

The Cabibbo angle : an algebraic calculation
Quentin Duret, Bruno Machet
To cite this version:
Quentin Duret, Bruno Machet. The Cabibbo angle : an algebraic calculation. The Cabibbo
angle can be positive or negative, which was overlooked in the first version. Minor a.. 2006.
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THE CABIBBO ANGLE : AN ALGEBRAIC CALCULATION
Q. Duret∗ and B. Machet†
Laboratoire de Physique The´orique et Hautes ´Energies, (Paris, France)‡
(Dated: September 11th 2006)
We show that the Cabibbo angle θc satisfies the relation tan(2θc) = ±1/2 when universality
for diagonal neutral currents of mass eigenstates is satisfied at the same level of accuracy as the
absence of their off-diagonal counterparts. The predicted value is cos θc ≈ 0.9732, only 7/10000
away from experimental results. No mass ratio appears in the calculation. θc occurs a priori for
both quark species, and, showing that one recovers the standard dependence of leptonic and semi-
leptonic decays of pseudoscalar mesons, we advocate, like for neutrinos and charged leptons, for a
symmetrical treatment of u and d-type quarks.
PACS numbers: 12.15.Ff 12.15.Hh 13.20.-v
INTRODUCTION
Why mixing angles are what they are is, in addition to their more and more precise experimental
determination, one of the most active present domain of research, both in the leptonic and hadronic
sectors. The attempts that have been proposed up to now in order to address this fundamental question,
such as the linking of the sine of the Cabibbo angle [1] to the d and s quark mass ratio [2], are often
based on empirical evidence, and so remain unsatisfying 1 . Indeed, there are actually too many free
parameters in the most general Yukawa couplings of the standard model to infer mixing angles from
mass ratios.
In [4], we have shown that maximal mixing angles naturally arise for non-degenerate coupled sys-
tems of physical particles with two generations, like neutrinos, when one requires universality for the
diagonal neutral currents of mass eigenstates and the absence of their off-diagonal counterparts. This
follows from an important feature that we also demonstrated there, i.e. that the mixing matrix of such
systems should never be parametrized as unitary. Such a property had also been checked before [5] for
neutral kaons.
In this letter, we would like to suggest one promising step on the path to a better understanding
of the enigma raised above. In direct continuation of [4], still working in the simple case of two
1 The rigorous treatment [3] shows that only weaker “asymptotic” relations hold, which involve both md/ms and mu/mc.
2generations, we investigate non-degenerate coupled systems that do not exist on-shell, like quarks.
Those systems, as shown in [4], are more naturally endowed with a unitary mixing matrix, due to the
fact that their two mixing angles finally satisfy θ2 = ±θ1[π]. Staying in an infinitesimal, but still
2-dimensional, neighborhood of such a “Cabibbo-case” (as referred to in [4]), we impose the physical
requirement that universality for weak diagonal neutral currents of mass eigenstates is realized with
the same accuracy as the absence of the off-diagonal mass changing neutral currents (MCNC’s). As a
consequence the value of the unique mixing angle is shown to be very close to that of the measured
Cabibbo angle.
GENERALITIES, STATES AND (WEAK) CURRENTS
In quantum field theory, the states of definite mass for a system of particles are defined as the
eigenstates, at its poles, of the full (renormalized) propagator, which is given by a matrix of dimension
2nf in flavour space. For non-degenerate coupled systems, like neutral kaons, leptons or quarks, these
mass eigenstates belong to different bases [4]. Indeed, one finds only one such eigenstate at each of
the poles of the propagator ; it is, among the 2nf eigenstates of the latter, the one corresponding to
the vanishing eigenvalue. Hence, the mass eigenstates do not make up an orthonormal basis, and the
mixing matrix, which by definition connects the set of mass eigenstates to that of flavour eigenstates,
is non-unitary.
Nevertheless, for systems of coupled fields which, like quarks, are never on-shell, a natural basis
appears : the one that occurs at any given z = q2. It is then orthonormal as soon as the inverse
propagator L(2)(z) is hermitian. Thus, the mixing matrix relating it to the flavour basis is unitary
(accorded that the flavour basis is orthonormal, too). This has, in the simple case of two generations,
the following consequences. While, in general, (2 × 2) mixing matrices for coupled systems are to
be parametrized with two mixing angles θ1 and θ2, quark-like (Cabibbo-like) systems finally shrink to
one dimension.
As stressed in [4], the physics of mixing angles is underlain by weak currents in mass space. These
we shall consider again now. Neutral (left-handed) weak currents for mass eigenstates are determined
by the combinations K†1K1 and K
†
2K2, where K1 and K2 are the mixing matrices for the two types of
fermions involved (for example neutral and charged leptons and, later, for our purpose, quarks of the
u and d-type), whereas charged currents give rise to the combination matrix K = K†1K2. As soon as
neither K1 nor K2 is unitary, the conditions for universality (equality of diagonal neutral currents) and
3absence of off-diagonal neutral currents, which are built-in properties in flavour space, appear on the
contrary no longer trivial in the space of mass eigenstates.
Let us parametrize like in [4] (c1 stands for example for cos θ1)
K1 =

 e
iαc1 e
iδs1
−eiβs2 eiγc2

 . (1)
It is then straightforward to show that universality in mass space for weak neutral currents imposes the
vanishing of
R = c21 + s
2
2 − c22 − s21, (2)
while the absence of MCNC’s requires that of
S = c1s1 − c2s2 or T = c1s1 + c2s2. (3)
GETTING THE CABIBBO ANGLE FROM AN EXTRA TOUCH OF PHYSICS
Let us now demand that the two conditions for universality and for the absence of MCNC’s are
realized with the same accuracy. We consider accordingly the equations
|R| = |S| ⇔ R± S = 0 or |R| = |T | ⇔ R± T = 0, (4)
i.e.
cos(2θ1)− cos(2θ2)± 1
2
(
sin(2θ1)− sin(2θ2)
)
= 0, (5a)
cos(2θ1)− cos(2θ2)± 1
2
(
sin(2θ1) + sin(2θ2)
)
= 0, (5b)
which directly lead to the following conditions :
sin(θ1 − θ2)
(
cos(θ1 + θ2)∓ 2 sin(θ1 + θ2)
)
= 0, (6a)
sin(θ1 + θ2)
(
cos(θ1 − θ2)∓ 2 sin(θ1 − θ2)
)
= 0. (6b)
First, one immediately sees that a trivial solution of the equations (6a) and (6b) is, resp., θ2 =
θ1 mod π and θ2 = −θ1 mod π. This corresponds to Cabibbo-like solutions, for which the two
4conditions R = 0 and T or S = 0 are separately satisfied, ensuring universality for diagonal neutral
currents in mass space, and the absence of MCNC’s.
Let us now expand (6a) and (6b) in the vicinity of these Cabibbo-like solutions by writing respec-
tively in there θ2 = ±θ1 + ǫ :
ǫ
((
cos(2θ1)∓ 2 sin(2θ1)
)− 2ǫ( sin(2θ1)± 2 cos(2θ1)
))
= 0, (7a)
ǫ
((
cos(2θ1)∓ 2 sin(2θ1)
)
+ 2ǫ
(
sin(2θ1)± 2 cos(2θ1)
))
= 0, (7b)
and impose that the two conditions (7a,7b) are satisfied with the same accuracy o(ǫ2). This leads to
the following equation (we replace hereafter θ1 with θc) :
tan(2θc) = ±1
2
, (8)
that can be rewritten
(
tan(θc)
)2 ± 4 tan(θc)− 1 = 0. (9)
Eqs. (9) admit the solutions : tan(θc) = ±(−2±
√
5), the smaller of which (they have opposite signs)
have the following value for their cosine : cos
(± arctan(2−√5)) ≃ 0.9732. This result lies remark-
ably close to the present experimental range [6] for the cosine of the Cabibbo angle : [0.9739, 0.9751].
The other solution corresponds to θc − π/2, i.e. to a simple relabeling of the two corresponding mass
states (for example um ↔ cm). It is noticeable that this approach yields a constant value for the
Cabibbo angle, which should be confronted with experiment. The most natural, a priori q2-dependent,
orthonormal basis related to Cabibbo-like systems mentioned in section 2, should then also exhibit
special properties with respect to its q2 dependence. This will be investigated in a forthcoming work.
A similar demonstration holds for K2.
PLEA FOR A SYMMETRICAL TREATMENT OF BOTH QUARK SPECIES
We already advocated in [4] for a symmetrical treatment of charged and neutral leptons, with max-
imal mixing for both species. The PMNS matrix then reduces to the diagonal unit matrix, without
putting in jeopardy the interpretation of the solar neutrino deficit on earth in terms of neutrino oscilla-
tions.
5We advocate for the same attitude in the quark sector. First, there is not a single good reason to
assume that mass and flavour eigenstates coincide for one species and not for the other one. Secondly,
the Cabibbo angle naturally arises by the mechanism explained above for both quark types. Last, it
is easy to demonstrate that, by a suitable analysis, we recover standard results. We give below the
example of semi-leptonic and leptonic decays of the K+ meson.
From the most common example of neutral kaons [7], two species of mesons are known to occur
in physics : flavor and mass eigenstates. The first are produced by strong interactions (which commute
with flavour) while the second also identify with electroweak eigenstates 2 . When studying its semi-
leptonic decays, the K+ is produced by strong interactions while electroweak eigenstates are detected
; the following subscripts should be accordingly attributed to this reaction : K+f → π0mℓ+mνem. In
a quark-spectator model, quark flavors are by definition weakly coupled with diag(1, 1) ; the decay
channels of K+f can only be, accordingly, (df s¯f ) ℓ+νℓ and (uf c¯f ) ℓ+νℓ, with coefficient 1 since no
mixing angle occurs. When projected on mass states, these final flavour states undergo the Cabibbo ro-
tations which take place for both quark species. This gives (df s¯f ) ≡ K0f = c2θcK0m+cθcsθc
(
(sms¯m)−
(dmd¯m)
)−s2θcK0m, (uf c¯f ) ≡ D0f = c2θcD0m+cθcsθc
(
(cmc¯m)−(umu¯m)
)−s2θcD0m. All semi-leptonic
decays of the K+f mesons are then described with their observed dependence on the Cabibbo angle (the
ones into π0, with their cos θc sin θc dependence, are built up from the uu¯ and dd¯ final states). We also
find double-Cabibbo-suppressed |∆S| = 2 transitions.
Leptonic decays are likewise easily explained by the decomposition K+f ≡ (uss¯f ) = c2θcK+m +
cθcsθc(D
+
sm − π+m)− s2θcD+m and with the leptonic decay of the π+m that takes place with a coefficient
1 since, now, no mixing angle occurs in charged weak currents.
This supports our point of view according to which the Cabibbo angle appears at the level of both
quark species but not in the charged weak couplings of the Lagrangian. While the usual “Cabibbo
matrix”, which connects mass eigenstates for u and d-type quarks, reduces then to the unit matrix, the
distinction between flavour and mass eigenstates for mesonic bound states must be carefully performed
3
.
2 This is not to be confused with the convention that flavor states of quarks and leptons are the ones which diagonally couple,
by definition, to weak gauge bosons.
3 Since the Cabibbo-angle keeps acting between flavour eigenstates of one type and mass eigenstates of the other type,
the evaluation of “box-diagram” controlling, for example, the transitions between the flavour K0 and K0 mesons stays
unchanged since internal quark lines are propagating, that is mass eigenstates, while external lines are the df and sf
flavour states.
6CONCLUSION AND PROSPECTS
Realizing, as was first done in [5], that mixing matrices of non-degenerate coupled systems should
not be parametrized as unitary, led in [4] to uncover maximal mixing of leptons as a class of solutions
of very simple physical conditions for their mass eigenstates.
Now, we have shown that the measured value of the Cabibbo angle θc is such that, in its vicinity
(allowing the second mixing angle θ2 to be close to θ1 = θc), universality of diagonal neutral currents
for mass eigenstates is realized with the same o
(
(θ2 − θ1)2
)
accuracy as the absence of MCNC’s.
In this elementary algebraic calculation, no mass ratio appears. It may thus help to provide indepen-
dent information on the latter. On another side, this feature is welcome for quark-like systems which
cannot be defined on-shell and for which, accordingly, the notion of physical mass is ill-defined.
Like for leptons, we advocate that the two species of quarks should be treated on equal footing.
Indeed, the Cabibbo angle, as a mixing angle, must occur for both, while it should not appear inside
charged weak couplings of mass eigenstates.
This work, together with [4], strongly suggests that the observed values of mixing angles for quarks
and leptons follow from well defined physical requirements. The generalization to three generations is
currently under investigation.
∗ E-mail: duret@lpthe.jussieu.fr
† E-mail: machet@lpthe.jussieu.fr
‡ UMR 7589 (CNRS / Universite´ Pierre et Marie Curie-Paris6 / Universite´ Denis Diderot-Paris7).
Postal address: LPTHE tour 24-25, 5e`me e´tage, Universite´ P. et M. Curie, BP 126, 4 place Jussieu, F-75252
Paris Cedex 05 (France)
[1] N. CABIBBO : Phys. Rev. Lett. 10 (1963) 531-533.
[2] S. WEINBERG : “The problem of mass”, in : A Festschrift for I.I. Rabi, Trans. N.Y. Acad. Sci. (Ser. II)
(1977) 185-201.
[3] B. MACHET & S.T. PETCOV : Phys. Lett. B 513 (2001) 371-380, and references therein.
[4] Q. DURET & B. MACHET : “Mixing Angles and Non-Degenerate Coupled Systems of Particles”, hep-
ph/0606303, and references therein.
[5] B. MACHET, V.A. NOVIKOV & M.I. VYSOTSKY : Int. J. Mod. Phys. A 20 (2005) 5399-5452, hep-
ph/0407268, and references therein.
[6] PARTICLE DATA GROUP : Review of Particle Physics, Phys. Lett. B 592 (2004) p.130.
7[7] See for example : R. BELU ˘SEVI ´C : “Neutral Kaons”, Springer Tracts in Modern Physics 153 (Springer-
Verlag Berlin Heidelberg New-York, 1999), p.8, and references therein.


The Cabibbo angle : an algebraic calculation

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